NAVIER_STOKES_2D_EXACT
Exact solutions to the
2D Incompressible Time-Dependent Navier Stokes Equations

NAVIER_STOKES_2D_EXACT, a FORTRAN77 library which evaluates exact solutions to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 2D.

• Lucas: steady flow, pressure is zero everywhere;
• Spiral: velocity is zero on the boundary of the unit square;
• Taylor: source term is zero everywhere.

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

NAVIER_STOKES_2D_EXACT is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

NAVIER_STOKES_3D_EXACT, a FORTRAN77 library which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 3D.

NAVIER_STOKES_MESH2D, MATLAB data files which define triangular meshes for several 2D test problems involving the Navier Stokes equations for fluid flow, provided by Leo Rebholz.

SPIRAL_DATA, a FORTRAN77 library which computes a velocity vector field that satisfies the continuity equation, writing the data to a file that can be plotted by gnuplot.

STOKES_2D_EXACT, a FORTRAN77 library which evaluates exact solutions to the incompressible steady Stokes equations over the unit square in 2D.

Reference:

1. Maxim Olshanskii, Leo Rebholz,
   Application of barycenter refined meshes in linear elasticity and incompressible fluid dynamics,
   ETNA: Electronic Transactions in Numerical Analysis,
   Volume 38, pages 258-274, 2011.
2. Geoffrey Taylor,
   On the decay of vortices in a viscous fluid,
   Philosophical Magazine,
   Volume 46, 1923, pages 671-674.
3. Geoffrey Taylor, A E Green,
   Mechanism for the production of small eddies from large ones,
   Proceedings of the Royal Society of London,
   Series A, Volume 158, 1937, pages 499-521.

Source Code:

• ns2de.f, the source code.
• ns2de.sh, BASH commands to compile the source code.

Examples and Tests:

• ns2de_prb.f, a sample calling program.
• ns2de_prb.sh, BASH commands to compile and run the sample program.
• ns2de_prb_output.txt, the output file.

Lucas Bystricky flow:

• lucas_commands.txt, commands to draw the velocity field.
• lucas_data.txt, velocity field data.
• lucas.png, a PNG plot of the velocity field.

Spiral flow:

• spiral_commands.txt, commands to draw the velocity field.
• spiral_data.txt, velocity field data.
• spiral.png, a PNG plot of the velocity field.

Taylor flow:

• taylor_commands.txt, commands to draw the velocity field.
• taylor_data.txt, velocity field data.
• taylor.png, a PNG plot of the velocity field.

List of Routines:

• GET_UNIT returns a free FORTRAN unit number.
• GRID_2D returns a regular 2D grid.
• NS2DE_GNUPLOT writes the Navier-Stokes velocity field to files for GNUPLOT.
• R8VEC_AMAX returns the maximum absolute value in an R8VEC.
• R8VEC_AMIN returns the minimum absolute value in an R8VEC.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• R8VEC_MAX returns the maximum value in an R8VEC.
• R8VEC_MIN returns the minimum value in an R8VEC.
• R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
• R8VEC_UNIFORM_AB returns a scaled pseudorandom R8VEC.
• RESID_LUCAS returns Lucas Bystricky residuals.
• RESID_SPIRAL evaluates the pointwise residual of the spiral flow problem.
• RESID_TAYLOR returns residuals of the Taylor exact Navier Stokes solution.
• RHS_LUCAS evaluates the right hand side of Lucas Bystricky's problem.
• RHS_SPIRAL evaluates the right hand side of the spiral flow problem.
• RHS_TAYLOR returns right hand sides of the Taylor vortex equations.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.
• UVP_LUCAS evaluates Lucas Bystricky's exact Navier Stokes solution.
• UVP_SPIRAL returns velocity and pressure for the spiral flow.
• UVP_TAYLOR evaluates the Taylor exact Navier Stokes solution.

You can go up one level to the FORTRAN77 source codes.